The most recent episode of WNYC’s Radiolab was about human limits. The first two stories were on the limits of the human body and mind and had people telling stories of surviving extreme endurance events and participating in memory competitions. The last story was about the limits of science. I was expecting the usual take on Godel’s Incompleteness Theorems but they touched on something different. Instead they talked about how this algorithm called Eureqa written by two Cornell computer scientists could deduce the dynamics of unknown systems. They used it to deduce the equations of a double pendulum based on the time series of the angles and angular velocities. They then applied it to a biological system and produced some dynamical equations. However, they then claimed that they had trouble publishing the results because they couldn’t explain what those equations described or meant. Steve Strogatz then came on and started to lament on the fact that as we begin to explore more and more complex systems, our brains may not be able to ever understand it. He basically said that once we reached that limit we may need to hand over science to computers.

I think that Steve is confounding the limits of a single human being with the limits of humans in general. To me, understanding is all about data compression. One says they understand something when they can give a simpler description of it or relate it to something they know already. Understanding is not a binary process. I understand some things better than other things and I attain a greater and sometimes lesser understanding of things the more I think about them. However, I do agree that there may be limits to the number of different things I personally can understand. This applies to things that other humans already understand like for example Turkish. Now, perhaps if I studied hard enough I could learn to speak Turkish but the time it took for me to do that would preclude me from learning something else like say Category Theory.

What Steve was specifically referring to I believe was that it may be difficult or impossible to understand certain complex systems by trying to relate them to what we know now. I think this is probably true but that doesn’t mean we won’t have intuitive understanding of such systems in the future. For example, it would be very difficult for an adult 500 years ago, who should be genetically indistinguishable to a person alive today, to understand Andrew Wile’s proof of Fermat’s last theorem. Most of them would first have to learn how to read and then learn 500 years of mathematics. Wiles basically used everything humans know about math up to this point to prove the theorem. The average mathematician alive today that doesn’t specialize in arithmetic algebraic geometry has trouble understanding the proof. They simply don’t have the background to follow all the arguments.

Now, I do believe that there are things that we can never understand because we are bound by the rules of computation. Turing showed us that there are undecidable problems that cannot be solved in general like if a computation will halt. However, I do think that humans have the capability to understand individual things that arise out of computations and that includes physical objects like biological systems. We may not know whether a given computation will halt but we could understand what has already been computed. For complex systems that Steve was alluding to, we just don’t yet know what the form of that understanding will be. Consider Brownian motion, which is the modern paradigm of an unpredictable process. Until Einstein pointed out that the process should be understood probabilistically and calculated the time dependence of the mean square deviation of a Brownian particle people didn’t even know how to think about the phenomenon. I think most physicists would claim that they have a good understanding of Brownian motion even though they have no idea what a single trajectory of a Brownian particle will do. From a neuroscience perspective, Brownian motion has become a primitive concept and we can understand more complex things in terms of it. I think this will hold true for even more complex phenomenon. We can always reform what we consider to be intuitive and build from that.

Addendum: I forgot to relate everything back to the title of the post. I called this the scale invariant life because I think everyone will go through similar stages where they learn what is known, make some new discoveries and then reach a crisis where they can’t understand something new in terms of what they already know. Thus there are no absolute thresholds of discovery or knowledge. We just make excursions from where we start and then the next generation takes over.